Numerical modeling of elastic waves in micropolar plates and shells taking into account inertial characteristics

Abstract

Mathematical models of micropolar plates and shells are considered within the framework of the approximation approach. The governing equations of the theories are written in a thermodynamically consistent form of the conservation laws. This ensures hyperbolicity and correctness of the initial boundary value problems. For numerical solution, we propose parallel algorithms for supercomputers with graphics processing units. The algorithms are based on the splitting method with respect to spatial variables. We present the results of numerical computations of wave propagation in micropolar rectangular plates and cylindrical panels for media with different types of microstructure particles.

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References

  1. 1.

    Abali, B.E., Müller, W.H., dell’Isola, F.: Theory and computation of higher gradient elasticity theories based on action principles. Arch. Appl. Mech. 87(9), 1495–1510 (2017)

    ADS  Article  Google Scholar 

  2. 2.

    Abali, B.E.: Revealing the physical insight of a length-scale parameter in metamaterials by exploiting the variational formulation. Contin. Mech. Thermodyn. (2018). https://doi.org/10.1007/s00161-018-0652-8

    ADS  MathSciNet  Article  Google Scholar 

  3. 3.

    Abd-Alla, A., Giorgio, I., Galantucci, L., Hamdan, A.M., Del Vescovo, D.: Wave reflection at a free interface in an anisotropic pyroelectric medium with nonclassical thermoelasticity. Contin. Mech. Thermodyn. 28(1–2), 67–84 (2016)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Altenbach, J., Altenbach, H., Eremeyev, V.A.: On generalized Cosserat-type theories of plates and shells. A short review and bibliography. Arch. Appl. Mech. 80(1), 73–92 (2010)

    ADS  MATH  Article  Google Scholar 

  5. 5.

    Altenbach, H., Eremeyev, V.A.: On the linear theory of micropolar plates. ZAMM 89(4), 242–256 (2009)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Altenbach, H., Eremeyev, V.A.: On the constitutive equations of viscoelastic micropolar plates and shells of differential type. Math. Mech. Complex Syst 3(3), 273–283 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Ambartsumian, S.A.: The theory of transverse bending of plates with asymmetric elasticity. Mech. Compos. Mater. 32(1), 30–38 (1996)

    ADS  Article  Google Scholar 

  8. 8.

    Bagdoev, A.G., Erofeyev, V.I., Shekoyan, A.V.: Wave Dynamics of Generalized Continua. Springer, Heidelberg (2016)

    Google Scholar 

  9. 9.

    Berezovski, A., Giorgio, I., Corte, A.D.: Interfaces in micromorphic materials: wave transmission and reflection with numerical simulations. Math. Mech. Solids 21(1), 37–51 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Bîrsan, M., Neff, P.: On the characterization of drilling rotation in the 6-parameter resultant shell theory. In: Pietraszkiewiecz, W. (ed.) Shell Structures: Theory and Applications, vol. 3. Taylor & Francis Group, London (2013)

    Google Scholar 

  11. 11.

    Cosserat, E., Cosserat, F.: Théorie des Corps Déformables. Herman et Fils, Paris (1909)

    Google Scholar 

  12. 12.

    dell’Isola, F., Madeo, A., Placidi, L.: Linear plane wave propagation and normal transmission and reflection at discontinuity surfaces in second gradient 3D continua. ZAMM 92(1), 52–71 (2012)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Del Vescovo, D., Giorgio, I.: Dynamic problems for metamaterials: review of existing models and ideas for further research. Int. J. Eng. Sci. 80, 153–172 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Eremeyev, V.A., Zubov, L.M.: On constitutive inequalities in nonlinear theory of elastic shells. ZAMM 87(2), 94–101 (2007)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Eremeyev, V.A., Zubov, L.M.: Mechanics of Elastic Shells (in Russ.). Nauka, Moscow (2008)

    Google Scholar 

  16. 16.

    Eremeyev, V.A., Lebedev, L.P., Altenbach, H.: Foundations of Micropolar Mechanics. Springer, Heidelberg (2013)

    Google Scholar 

  17. 17.

    Ericksen, J.L., Truesdell, C.: Exact theory of stress and strain in rods and shells. Arch. Ration. Mech. Anal. 1(1), 295–323 (1958)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Eringen, A.C.: Theory of micropolar plates. ZAMP 18(1), 12–30 (1967)

    ADS  Google Scholar 

  19. 19.

    Eringen, A.C.: Microcontinuum Field Theory I. Foundations and Solids. Springer, New York (1999)

    Google Scholar 

  20. 20.

    Erofeyev, V.I.: Wave Processes in Solids with Microstructure. World Scientific, London (2003)

    Google Scholar 

  21. 21.

    Farber, R.: CUDA Application Design and Development. Morgan Kaufmann, Burlington (2011)

    Google Scholar 

  22. 22.

    Forest, S., Sab, K.: Finite-deformation second-order micromorphic theory and its relations to strain and stress gradient models. Math. Mech. Solids (2017). https://doi.org/10.1177/1081286517720844

  23. 23.

    Friedrichs, K.O.: Symmetric hyperbolic linear differential equations. Commun. Pure Appl. Math. 7(2), 345–392 (1954)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Green, A.E., Naghdi, P.M.: Linear theory of an elastic Cosserat plate. Camb. Phil. Soc. Math. Phys. Sci. 63(2), 537–550 (1967)

    MATH  Article  Google Scholar 

  25. 25.

    Ivanova, E.A., Vilchevskaya, E.N.: Micropolar continuum in spatial description. Contin. Mech. Thermodyn. 28, 1759–1780 (2016)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Kulikovskii, A.G., Pogorelov, N.V., Semenov, A.Y.: Mathematical Aspects of Numerical Solution of Hyperbolic Systems, Monographs and Surveys in Pure and Applied Mathematics, vol. 118. Chapman & Hall, Boca Raton (2001)

    Google Scholar 

  27. 27.

    Lakes, R.S.: Experimental methods for study of Cosserat elastic solids and other generalized continua. In: Muhlhaus, H. (ed.) Continuum models for materials with micro-structure. J. Wiley, New York, Ch. 1, pp. 1–22 (1995)

  28. 28.

    Lebedev, L.P., Cloud, M.J., Eremeyev, V.A.: Tensor Analysis with Applications in Mechanics. World Scientific, New Jersey (2010)

    Google Scholar 

  29. 29.

    Madeo, A., Neff, P., Ghiba, I.D., Placidi, L., Rosi, G.: Wave propagation in relaxed micromorphic continua: modeling metamaterials with frequency band-gaps. Contin. Mech. Thermodyn. 27(4–5), 551–570 (2015)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Marchuk, G.I.: Methods of Numerical Mathematics. Springer, Berlin (1975)

    Google Scholar 

  31. 31.

    Marchuk, G.I.: Splitting Methods (in Russ.). Nauka, Moscow (1988)

    Google Scholar 

  32. 32.

    Neff, P., Ghiba, I.D., Madeo, A., Placidi, L., Rosi, G.: A unifying perspective: the relaxed linear micromorphic continuum. Contin. Mech. Thermodyn. 26(5), 639–681 (2014)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Polizzotto, C.: A second strain gradient elasticity theory with second velocity gradient inertia—Part I: constitutive equations and quasi-static behavior. Int. J. Solids Struct. 50(24), 3749–3765 (2013)

    Article  Google Scholar 

  34. 34.

    Polizzotto, C.: A second strain gradient elasticity theory with second velocity gradient inertia—Part II: dynamic behavior. Int. J. Solids Struct. 50(24), 3766–3777 (2013)

    Article  Google Scholar 

  35. 35.

    Reissner, E.: A note on generating generalized two-dimensional plate and shell theories. Z. Angew. Math. Phys. 28, 633–642 (1977)

    MATH  Article  Google Scholar 

  36. 36.

    Sadovskii, V.M., Sadovskaya, O.V., Varygina, M.P.: Numerical modeling of three-dimensional wave motions in couple-stress media (In Russ.). Comput. Contin. Mech. 2(4), 111–121 (2009)

    Article  Google Scholar 

  37. 37.

    Sadovskaya, O., Sadovskii, V., Varygina, M.: Numerical solution of dynamic problems in couple-stressed continuum on multiprocessor computer systems. Int. J. Num. Anal. Model. Ser. B 2(2–3), 215–230 (2011)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Sadovskaya, O., Sadovskii, V.: Mathematical modeling in mechanics of granular materials. In: Altenbach, H. (ed.) Series of Advanced Structured Materials, vol. 21. Springer, Heidelberg (2012)

    Google Scholar 

  39. 39.

    Sargsyan, S.O.: The theory of micropolar thin elastic shells. J. Appl. Math. Mech. 76, 235–249 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  40. 40.

    Sarkisyan, S.O.: Mathematical model of micropolar elastic thin plates and their strength and stiffness characteristics. J. Appl. Mech. Tech. Phys. 53(2), 275–282 (2012)

    ADS  MATH  Article  Google Scholar 

  41. 41.

    Steinberg, L., Kvasov, R.: Enhanced mathematical model for Cosserat plate bending. Thin-Walled Struct. 63, 51–62 (2013)

    Article  Google Scholar 

  42. 42.

    Shared Facility Center “Data Center of FEB RAS” (Khabarovsk). http://lits.ccfebras.ru

  43. 43.

    Varygina, M., Sadovskaya, O., Sadovskii, V.: Resonant properties of moment Cosserat continuum. J. Appl. Mech. Tech. Phys. 51(3), 405–413 (2010)

    ADS  MATH  Article  Google Scholar 

  44. 44.

    Varygina, M.: Numerical modeling of wave propagation processes in micropolar rods and thin plates. AIP Conf. Proc. 1773, 08007-1–08007-8 (2016)

    Google Scholar 

  45. 45.

    Varygina, M.: Numerical modeling of micropolar thin elastic plates. LNCS 10187, 690–697 (2017)

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Varygina, M.: Numerical modeling of micropolar cylindrical shells on supercomputers with GPUs. AIP Conf. Proc. 1895, 080005-1–080005-8 (2017)

    Google Scholar 

  47. 47.

    Varygina, M.: Computer simulation of cylindrical shell deformation based on micropolar media equations. In: Pietraszkiewicz, W., Witkowski, W. (eds.) Shell Structures: Theory and Applications, vol. 4, pp. 395–398. Taylor & Francis, London (2017)

    Google Scholar 

  48. 48.

    Yanenko, N.N.: The Method of Fractional Steps. The Solution of Problems of Mathematical Physics in Several Variables. Springer, Berlin (1971)

    Google Scholar 

  49. 49.

    Wilson, E.B.: Vector Analysis, Founded Upon the Lectures of J.W. Gibbs. Yale University Press, New Haven (1901)

    Google Scholar 

  50. 50.

    Yang, W.H.: A useful theorem for constructing convex yield functions. Trans. ASME J. Appl. Mech. 47(2), 301–305 (1980)

    ADS  MathSciNet  MATH  Article  Google Scholar 

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Acknowledgements

This work was supported by the Russian Foundation for Basic Research Grant 18-31-00100.

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Correspondence to Maria Varygina.

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Communicated by Dr. Francesco dell’Isola.

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Varygina, M. Numerical modeling of elastic waves in micropolar plates and shells taking into account inertial characteristics. Continuum Mech. Thermodyn. 32, 761–774 (2020). https://doi.org/10.1007/s00161-018-0725-8

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Keywords

  • Micropolar plates and shells
  • Inertia tensor
  • Dynamic problems
  • Wave propagation
  • Parallel computations